Algebra Problem (Math)!

* Write the expression in the form a + bi: i ^ 251 - i ^ 38

Solve the equations with radical and exponents: √x+7 - 2√x = -2


Thank you for your help.

We know that i^2 = -1 and hence i^4 = 1.

Divide i^4 into i^251 and you get ((i^4)^62) * i^3 = 1 * i^3 = -i

i^38 = ((i^4)^9) * i^2 = 1 * i^2 = -1

-i - (-1) = 1 - i <------------ Answer to first problem

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Square both sides, getting

x + 7 + 4x - (4 * √(x+7) * √x) = 4

Putting the radicals on the right and the non-radicals on the left, we get:

- (4 * √(x+7) * √x) = -x - 3...............Multiply both sides by (-1)

(4 * √(x+7) * √x) = x + 3............Since (4 * √(x+7) * √x) = 4 * √(x+7) * √x =

4 * √(x^2 + 7x) put that one right side.

4 * √(x^2 + 7x) = (x + 3)................square both sides

16x^2 + 112x = x^2 + 6x + 9

15x^2 + 106x - 9 = 0

the determinant b^2 - 4ac is 11,236 + 540 = 11,776 factoring is going to be difficult to impossible. So we fall back on the quadratic formula.

For answers we get 0.083909 and -7.150576

But the problem started out with √x so we must discard the negative result as in the real numbers we cannot take the square root of a negative number.

Thus the only answer is x = 0.083909

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